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1 In this chapter we present Kharitonov's Theorem on robust Hurwitz stability of interval polynomials, dealing with both the real and complex cases This elegant result forms the basis for many of the results, to be developed later in the book, on robustness under parametric uncertainty We develop an important extremal property of the associated Kharitonov polynomials and give an application of this theorem to state feedback stabilization An extension of Kharitonov's Theorem to nested families of interval polynomials is described Robust Schur stability of interval polynomials is also discussed and it is shown that robust stability can be ascertained from that of the upper edges 51 INTRODUCTION We devote this chapter mainly to a result proved in 1978 by V L Kharitonov, regarding the Hurwitz stability of a family of interval polynomials This result was so surprising and elegant that it has been the starting point of a renewed interest in robust control theory with an emphasis on deterministic bounded parameter perturbations It is important therefore that control engineers thoroughly understand both result and proof, and this is why we considerably extend our discussion of this subject In the next section we rst state and prove Kharitonov's Theorem for real polynomials We emphasize how this theorem generalizes the Hermite-Biehler Interlacing Theorem which is valid for a single polynomial This has an appealing frequency domain interpretation in terms of interlacing of frequency bands We then give an interpretation of Kharitonov's Theorem based on the evolution of the complex plane image set of the interval polynomial family Here again the Boundary Crossing Theorem and the monotonic phase increase property of Hurwitz polynomials (Chapter 1) are the key concepts that are needed to establish the Theorem This proof gives useful geometric insight into the working of the theorem and shows how the result is related to the Vertex Lemma (Chapter ) We then state the theorem for the 3

2 4 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 case of an interval family of polynomials with complex coecients This proof follows quite naturally from the above interlacing point of view Next, we develop an important extremal property of the Kharitonov polynomials This property establishes that one of the four points represented by the Kharitonov polynomials is the closest to instability over the entire set of uncertain parameters The latter result is independent of the norm used to measure the distance between polynomials in the coecient space We next give an application of the Kharitonov polynomials to robust state feedback stabilization Following this we establish that Kharitonov's Theorem can be extended to nested families of interval polynomials which are neither interval or polytopic and in fact includes nonlinear dependence on uncertain parameters In the last section we consider the robust Schur stability of an interval polynomial family A stability test for this family is derived based on the upper edges which form a subset of all the exposed edges We illustrate the application of these fundamental results to control systems with examples 5 KHARITONOV'S THEOREM FOR REAL POLYNOMIALS In this chapter stable will mean Hurwitzstable unless otherwise stated Of course, we will say that a set of polynomials is stable if and only if each and every element of the set is a Hurwitzpolynomial Consider now the set I(s) of real polynomials of degree n of the form (s) = o + 1 s + s + 3 s s n s n where the coecients lie within given ranges, [x ;y ]; 1 [x 1 ;y 1 ]; 111; n [x n ;y n ]: Write := [ ; 1 ; 111; n ] and identify a polynomial (s) with its coecient vector Introduce the hyperrectangle or box of coecients 1 := 8 : IR n+1 ; x i i y i ; i =; 1; 111;n 9 : (5:1) We assume that the degree remains invariant over the family, so that = [x n ;y n ] Such a set of polynomials is called a real interval family and we loosely refer to I(s) as an interval polynomial Kharitonov's Theorem provides a surprisingly simple necessary and sucient condition for the Hurwitzstability of the entire family Theorem 51 (Kharitonov's Theorem) Every polynomial in the family I(s) is Hurwitz if and only if the following four extreme polynomials are Hurwitz: K 1 (s) =x o + x 1 s + y s + y 3 s 3 + x 4 s 4 + x 5 s 5 + y 6 s ;

3 Sec 5 KHARITONOV'S THEOREM FOR REAL POLYNOMIALS 5 K (s) =x o + y 1 s + y s + x 3 s 3 + x 4 s 4 + y 5 s 5 + y 6 s ; K 3 (s) =y o + x 1 s + x s + y 3 s 3 + y 4 s 4 + x 5 s 5 + x 6 s ; (5) K 4 (s) =y o + y 1 s + x s + x 3 s 3 + y 4 s 4 + y 5 s 5 + x 6 s : The box 1 and the vertices corresponding to the Kharitonov polynomials are shown in Figure 51 The proof that follows allows for the interpretation of Kharitonov's 6 1 K 1 t K t? 9 1 t K 3 t K 4 - Figure 51 The box 1 and the four Kharitonov vertices Theorem as a generalization ofthe Hermite-Biehler Theorem for Hurwitz polynomials, proved in Theorem 17 ofchapter 1 We start by introducing two symmetric lemmas that will lead us naturally to the proofofthe theorem Lemma 51 Let P 1 (s) =P even (s)+p odd (s) 1 P (s) =P even (s)+p odd (s) denote two stable polynomials of the same degree with the same even part P even (s) and diering odd parts P odd (s) and P odd (s) satisfying 1 P o 1 (!) P o (!); for all! [; 1]: (5:3)

4 6 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 Then, P (s) =P even (s)+p odd (s) is stable for every polynomial P (s) with odd part P odd (s) satisfying P o 1 (!) P o (!) P o (!); for all! [; 1]: (5:4) Proof Since P 1 (s) and P (s) are stable, P o (!) and P o (!) both satisfy the interlacing property with P e (!) In particular, P o (!) and P o (!) are not only of the 1 1 same degree, but the sign of their highest coecient is also the same since it is in fact the same as that of the highest coecient of P e (!) Given this it is easy to see that P o (!) cannot satisfy (54) unless it also has this same degree and the same sign for its highest coecient Then, the condition in (54) forces the roots of P o (!) to interlace with those of P e (!) Therefore, according to the Hermite-Biehler Theorem (Theorem 17, Chapter 1), P even (s)+p odd (s) is stable We remark that Lemma 51 as well as its dual, Lemma 5 given below, are special cases of the Vertex Lemma, developed in Chapter and follow immediately from it We illustrate Lemma 51 in the example below (see Figure 5) Example 51 Let P 1 (s) =s 7 +9s 6 +31s 5 +71s s s +76s +1 P (s) =s 7 +9s 6 +34s 5 +71s s s +83s +1: Then P even (s) =9s 6 +71s s +1 P odd 1 (s) =s 7 +31s s 3 +76s P odd (s) =s 7 +34s s 3 +83s: Figure 5 shows that P e (!) and the tube bounded by P o 1 (!) and P o (!) satisfy the interlacing property Thus, we conclude that every polynomial P (s) with odd part P odd (s) satisfying P o 1 (!) P o (!) P o (!); for all! [; 1] is stable For example, the dotted line shown inside the tube represents The dual of Lemma 51 is: Lemma 5 Let P odd (s) =s 7 +3s s 3 +79s: P 1 (s) =P even (s)+p odd (s) 1 P (s) =P even (s)+p odd (s)

5 Sec 5 KHARITONOV'S THEOREM FOR REAL POLYNOMIALS P o (!) rad/sec P o (!)! P o 1 (!) Figure 5 P e (!) and (P o 1 (!);P o (!)) (Example 51) P e (!) denote two stable polynomials of the same degree with the same odd part P odd (s) and diering even parts P even 1 (s) and P even (s) satisfying P e(!) P e (!); for all! [; 1]: (5:5) 1 Then, P (s) =P even (s)+p odd (s) is stable for every polynomial P (s) with even part P even (s) satisfying P e(!) P e (!) P e (!); for all! [; 1]: (5:6) 1 We are now ready to prove Kharitonov's Theorem Proof of Kharitonov's Theorem The Kharitonov polynomialsrepeated below, for convenience are four specic vertices of the box 1: K 1 (s) =x o + x 1 s + y s + y 3 s 3 + x 4 s 4 + x 5 s 5 + y 6 s ; K (s) =x o + y 1 s + y s + x 3 s 3 + x 4 s 4 + y 5 s 5 + y 6 s ;

6 8 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 K 3 (s) =y o + x 1 s + x s + y 3 s 3 + y 4 s 4 + x 5 s 5 + x 6 s ; (57) K 4 (s) =y o + y 1 s + x s + x 3 s 3 + y 4 s 4 + y 5 s 5 + x 6 s : These polynomials are built from two dierent even parts Kmax even (s) and Keven min (s) and two dierent odd parts Kmax odd (s) and Kodd min (s) dened below: and K even max (s) :=y o + x s + y 4 s 4 + x 6 s 6 + y 8 s ; K even min (s) :=x o + y s + x 4 s 4 + y 6 s 6 + x 8 s ; K odd max (s) :=y 1s + x 3 s 3 + y 5 s 5 + x 7 s 7 + y 9 s ; K odd min (s) :=x 1s + y 3 s 3 + x 5 s 5 + y 7 s 7 + x 9 s : The Kharitonov polynomials in (5) or (57) can be rewritten as: K 1 (s) =K even min (s) +Kodd min (s); K (s) =K even min (s) +Kodd max (s); K 3 (s) =K even max (s) +Kodd min (s); (58) K 4 (s) =K even max (s) +Kodd max (s): The motivation for the subscripts \max" and \min" is as follows Let (s) be an arbitrary polynomial with its coecients lying in the box 1 and let even (s) be its even part Then so that and K e max (!) =y x! + y 4! 4 x 6! 6 + y 8! ; e (!) =! + 4! 4 6! 6 + 8! ; K e min (!) =x y! + x 4! 4 y 6! 6 + x 8! ; K e max (!) e (!) =(y )+( x )! +(y 4 4 )w 4 +( 6 x 6 )! ; e (!) K e min (!) =( x )+(y )! +( 4 x 4 )! 4 +(y 6 6 )! : Therefore, K e min (!) e (!) K e max (!); for all! [; 1]: (5:9) Similarly, if odd (s) denotes the odd part of (s), and odd (j!) =j! o (!) itcanbe veried that K o min (!) o (!) Kmax o (!); for all! [; 1]: (5:1)

7 Sec 5 KHARITONOV'S THEOREM FOR REAL POLYNOMIALS 9 Imag 6 K o max(!) K o min (!) I(j!) K e min (!) Ke max(!) Real - Figure 53 Axis parallel rectangle I(j!) Thus (j!) lies in an axis parallel rectangle I(j!) as shown in Figure 53 To proceed with the proof of Kharitonov's Theorem we note that necessityof the condition is trivial since if all the polynomials with coecients in the box 1 are stable, it is clear that the Kharitonov polynomials must also be stable since their coecients lie in 1 For the converse, assume that the Kharitonov polynomials are stable, and let (s) = even (s)+ odd (s) bean arbitrary polynomial belonging to the family I(s), with even part even (s) and odd part odd (s) We conclude, from Lemma 51 applied to the stable polynomials K 3 (s) and K 4 (s) in (58), that K even max (s)+ odd (s) is stable: (5:11) Similarly, from Lemma 51 applied to the stable poly nomials K 1 (s) and K (s) in (58) we conclude that K even min (s)+ odd (s) is stable: (5:1) Now, since (59) holds, we can applylemma 5 to the two stable polynomials K even max (s)+ odd (s) and K even min (s)+ odd (s) to conclude that even (s)+ odd (s) =(s) is stable: Since (s) was an arbitrarypolynomial of I(s) we conclude that the entire family of polynomials I(s) is stable and this concludes the proof of the theorem Remark 51 The Kharitonov polynomials can also be written with the highest order coecient as the rst term: ^K 1 (s) =xns n + yn1s n1 + yns n + xn3s n3 + xn4s n ;

8 3 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 ^K (s) =xns n + xn1s n1 + yns n + yn3s n3 + xn4s n ; ^K 3 (s) =yns n + xn1s n1 + xns n + yn3s n3 + yn4s n ; (513) ^K 4 (s) =yns n + yn1s n1 + xns n + xn3s n3 + yn4s n : Remark 5 The assumption regarding invariant degree of the interval family can be relaxed In this case some additional polynomials need to be tested for stability This is dealt with in Exercise 513 Remark 53 The assumption inherent in Kharitonov's Theorem that the coef- cients perturb independently is crucial to the working of the theorem In the examples belowwe have constructed some control problems where this assumption is satised Obviously in many real world problems this assumption would fail to hold, since the characteristic polynomial coecients would perturb interdependently through other primary parameters However even in these cases Kharitonov's Theorem can give useful and computationally simple answers by overbounding the actual perturbations by an axis parallel box 1 in coecient space Remark 54 As remarked above Kharitonov's Theorem would give conservative results when the characteristic polynomial coecients perturb interdependently The Edge Theorem and the Generalized Kharitonov Theorem described in Chapters 6 and 7 respectively were developed precisely to deal nonconservatively with such dependencies Example 5 Consider the problem of checking the robust stability of the feedback system shown in Figure 54 - j - s G(s) Figure 54 Feedback system (Example 5) The plant transfer function is G(s) = 1 s + s ( 4 s + 3 s 3 + ) with coecients being bounded as 4 [x 4 ;y 4 ]; 3 [x 3 ;y 3 ]; [x ;y ]; 1 [x 1 ;y 1 ]; [x ;y ]:

9 Sec 5 KHARITONOV'S THEOREM FOR REAL POLYNOMIALS 31 The characteristic polynomial of the family is written as (s) = 4 s s 3 + s + 1 s + : The associated even and odd polynomials for Kharitonov's test are as follows: K even min (s) =x + y s + x 4 s 4 ; K odd min (s) =x 1 s + y 3 s 3 ; The Kharitonov polynomials are: K even max (s) =y + x s + y 4 s 4 ; K odd max(s) =y 1 s + x 3 s 3 : K 1 (s) =x + x 1 s + y s + y 3 s 3 + x 4 s 4 ; K (s) =x + y 1 s + y s + x 3 s 3 + x 4 s 4 ; K 3 (s) =y + x 1 s + x s + y 3 s 3 + y 4 s 4 ; K 4 (s) =y + y 1 s + x s + x 3 s 3 + y 4 s 4 : The problem of checking the Hurwitz stability of the family therefore is reduced to that of checking the Hurwitz stability of these four polynomialsthis in turn reduces to checking that the coecients have the same sign (positive, say; otherwise multiply (s) by -1) and that the following inequalities hold: K 1 (s) Hurwitz : y y 3 >x 1 x 4 ; x 1 y y 3 >x 1 x 4 + y 3 x ; K (s) Hurwitz : y x 3 >y 1 x 4 ; y 1 y x 3 >y 1 x 4 + x 3 x ; K 3 (s) Hurwitz : x y 3 >x 1 y 4 ; x 1 x y 3 >x 1 y 4 + y 3 y ; K 4 (s) Hurwitz : x x 3 >y 1 y 4 ; y 1 x x 3 >y 1 y 4 + x 3 y : Example 53 Consider the control system shown in Figure 55 - j - C(s) - G(s) s Figure 55 Feedback system with controller (Example 53) The plant is described by the rational transfer function G(s) with numerator and denominator coecients varying independently in prescribed intervalswe refer to such a family of transfer functions G(s) asaninterval plantin the present example we take G(s) := G(s) = n s + n 1 s + n : s 3 + d s + d 1 s + d n [1; :5]; n 1 [1; 6]; n [1; 7]; d [1; 1]; d 1 [:5; 1:5]; d [1; 1:5] :

10 3 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 The controller is a constant gain, C(s) = k that is to be adjusted, if possible, to robustly stabilize the closed loop system More precisely we are interested in determining the range of values of the gain k [1; +1]for which the closed loop system is robustly stable, ie stable for all G(s) G(s) The characteristic polynomial of the closed loop system is: (k; s) =s 3 +(d + kn ) {z } (k) s +(d + kn ) s +(d 1 + kn 1 ) 1 (k) {z {z } } (k) : Since the parameters d i, n j, i =; 1;, j =; 1; vary independently it follows that for each xed k, (k; s) is an interval polynomial Using the bounds given to describe the family G(s) we get the following coecient bounds for positive k: (k) [1+k; 1+7k]; 1 (k) [:5+k; 1:5+6k]; (k) [1+k; 1:5+:5k]: Since the leading coecient is +1 the remaining coecients must be all positive for the polynomial to be Hurwitz This leads to the constraints: (a) 1+k>; :5+k>; 1+k>: From Kharitonov's Theorem applied to third order interval polynomials it can be easily shown that to ascertain the Hurwitz stability of the entire family it suces to check in addition to positivity of the coecients, the Hurwitz stability of only the third Kharitonov polynomial K 3 (s) In this example we therefore have that the entire system is Hurwitz if and only if in addition to the above constraints (a) we have: (:5+k)(1+k) > 1:5+:5k: From this it follows that the closed loop system is robustly stabilized if and only if k ( + p 5; +1]: To complete our treatment of this important result we give Kharitonov's Theorem for polynomials with complex coecients in the next section 53 KHARITONOV'S THEOREM FOR COMPLEX POLYNOMI- ALS Consider the set I 3 (s) of all complex polynomials of the form, (s) =( + j )+( 1 + j 1 )s ( n + j n )s n (5:14) with [x ;y ]; 1 [x 1 ;y 1 ]; 111; n [x n ;y n ](5:15)

11 Sec 53 KHARITONOV'S THEOREM FOR COMPLEX POLYNOMIALS 33 and [u ;v ]; 1 [u 1 ;v 1 ]; 111; n [un;vn]: (5:16) This is a complex interval family of polynomials of degree n which includes the real interval family studied earlier as a special case It is natural to consider the generalization of Kharitonov's Theorem for the real case to this family The Hurwitz stability of complex interval families will also arise naturally in studying the extremal H1 norms of interval systems in Chapter 9 Complex polynomials also arise in the study of phase margins of control systems and in time-delay systems Kharitonov extended his result for the real case to the above case of complex interval families We assume as before that the degree remains invariant over the family Introduce two sets of complex polynomials as follows: and K + 1 (s) :=(x + ju )+(x 1 + jv 1 )s +(y + jv )s +(y 3 + ju 3 )s 3 +(x 4 + ju 4 )s 4 +(x 5 + jv 5 )s ; K + (s) :=(x + jv )+(y 1 + jv 1 )s +(y + ju )s +(x 3 + ju 3 )s 3 +(x 4 + jv 4 )s 4 +(y 5 + jv 5 )s ; (517) K + (s) 3 :=(y + ju )+(x 1 + ju 1 )s +(x + jv )s +(y 3 + jv 3 )s 3 +(y 4 + ju 4 )s 4 +(x 5 + u 5 )s ; K + (s) 4 :=(y + jv )+(y 1 + ju 1 )s +(x + ju )s +(x 3 + jv 3 )s 3 +(y 4 + jv 4 )s 4 +(y 5 + ju 5 )s ; K 1 (s) :=(x + ju )+(y 1 + ju 1 )s +(y + jv )s +(x 3 + jv 3 )s 3 +(x 4 + ju 4 )s 4 +(y 5 + ju 5 )s ; K (s) :=(x + jv )+(x 1 + ju 1 )s +(y + ju )s +(y 3 + jv 3 )s 3 +(x 4 + jv 4 )s 4 +(x 5 + ju 5 )s ; K (s) 3 :=(y + ju )+(y 1 + jv 1 )s +(x + jv )s +(x 3 + ju 3 )s 3 (518) +(y 4 + ju 4 )s 4 +(y 5 + jv 5 )s ; K (s) 4 :=(y + jv )+(x 1 + jv 1 )s +(x + ju )s +(y 3 + ju 3 )s 3 +(y 4 + jv 4 )s 4 +(x 5 + jv 5 )s : Theorem 5 The family of polynomials I 3 (s) is Hurwitz if and only if the eight Kharitonov polynomials K + 1 (s), K+ (s), K+ 3 (s), K+ 4 (s), K 1 (s), K (s), K 3 (s), K 4 (s) are all Hurwitz Proof The necessity of the condition is obvious because the eight Kharitonov polynomials are in I 3 (s) The proof of suciency follows again from the Hermite- Biehler Theorem for complex polynomials (Theorem 18, Chapter 1)

12 34 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 Observe that the Kharitonov polynomials in (517) and (518) are composed of the following extremal polynomials: For the \positive" Kharitonov polynomials dene: R + max (s) :=y + ju 1 s + x s + jv 3 s 3 + y 4 s R + min (s) :=x + jv 1 s + y s + ju 3 s 3 + x 4 s I + max (s) :=jv + y 1 s + ju s + x 3 s 3 + jv 4 s I + min (s) :=ju + x 1 s + jv s + y 3 s 3 + ju 4 s so that K + 1 (s) =R+ min (s) +I+ min (s) K + (s) =R+ min (s) +I+ max (s) K + 3 (s) =R+ max (s) +I+ min (s) K + 4 (s) =R+ max (s) +I+ max (s): For the \negative" Kharitonov polynomials we have R max (s) =y + jv 1 s + x s + ju 3 s 3 + y 4 s R min (s) =x + ju 1 s + y s + jv 3 s 3 + x 4 s I max (s) =jv + x 1 s + ju s + y 3 s 3 + jv 4 s I min (s) =ju + y 1 s + jv s + x 3 s 3 + ju 4 s and K 1 (s) =R min (s) +I min (s) K (s) =R min (s) +I max (s) K 3 (s) =R max (s) +I min (s) K 4 (s) =R max (s) +I max (s): R 6 max (j!) and R6 min (j!) are real and I6 max (j!) and I6 min (j!) are imaginary Let Re[(j!)] := r (!) and Im[(j!)] := i (!) denote the real and imaginary parts of (s) evaluated at s = j! Then we have: r (!) = 1!! + 3! ; i (!) = + 1!! 3! : It is easy to verify that 8 < : 8 < : R + min (j!) r (!) R + max (j!); for all! [; 1] I + min (j!) j i (!) I+ max (j!) ; for all! [; 1] j R min (j!) r (!) R max (j!); for all! [; 1] I min (j!) j i (!) I max (j!) ; for all! [; 1] j (519) (5)

13 Sec 54 INTERLACING AND IMAGE SET INTERPRETATION 35 The proof of the theorem is now completed as follows The stability of the 4 positive Kharitonov polynomials guarantees interlacing of the \real tube" (bounded by R + max(j!) and R + (j!)) with the \imaginary tube" (bounded by min I+ max(j!) and I + min (j!)) for! The relation in (519) then guarantees that the real and imaginary parts of an arbitrary polynomial in I 3 (s) are forced to interlace for! Analogous arguments, using the bounds in (5) and the \negative" Kharitonov polynomials forces interlacing for! Thus by the Hermite-Biehler Theorem for complex polynomials (s) is Hurwitz Since (s) was arbitrary, it follows that each and every polynomial in I 3 (s) is Hurwitz Remark 55 In the complex case the real and imaginary parts of (j!) are polynomials in! and not!, and therefore it is necessary to verify the interlacing of the roots on the entire imaginary axis and not only on its positive part This is the reason why there are twice as many polynomials to check in the complex case 54 INTERLACING AND IMAGE SET INTERPRETATION In this section we interpret Kharitonov's Theorem in terms of the interlacing property or Hermite-Biehler Theorem and also in terms of the complex plane image of the set of polynomials I(s), evaluated at s = j! for each! [; 1] In Chapter 1 we have seen that the Hurwitz stability of a single polynomial (s) = even (s)+ odd (s)is equivalent to the interlacing property of e (!) = even (j!) and o (!) = odd (j!) j! In considering the Hurwitz stability of the interval family I(s) we see that the family is stable if and only if every element satises the interlacing property In view of Kharitonov's Theorem it must therefore be true that verication of the interlacing property for the four Kharitonov polynomials guarantees the interlacing property of every member of the family the following version of Kharitonov's Theorem Let! e max i itive roots of Kmax(!)(K e min e (!)) and let!max o i K o max(!)(k o min (!)) This point of view is expressed in (! e min i ) denote the pos- (! o min i ) denote the positive roots of Theorem 53 (Interlacing Statement of Kharitonov's Theorem) The family I(s) contains only stable polynomials if and only if 1) The polynomials Kmax(!), e Kmin e (!), Ko max(!), Kmin o (!) have only real roots and the set of positive roots interlace as follows: <! min e 1 <! max e 1 <! min o 1 <! max o 1 <! min e <! max e <! min o <! max o < 111; ) Kmax(), e Kmin e (), Ko max(), Kmin o () are non zero and of the same sign This theorem is illustrated in Figure 56 which shows how the interlacing of the odd and even tubes implies the interlacing of the odd and even parts of each polynomial in the interval family We illustrate this interlacing property of interval polynomials with an example

14 36 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 K e max(!) Kmax(!) o #! min o 1! max o 1! min e 1 B 1! max e 1 B! K e min (!)! " K o min (!) Figure 56 Interlacing odd and even tubes Example 54 Consider the interval family (s) =s s s s s 3 + s + 1 s + where 6 [9; 9:5]; 5 [31; 31:5]; 4 [71; 71:5]; 3 [111; 111:5]; [19; 19:5]; 1 [76; 76:5]; [1; 1:5] Then Kmax(!) e =9! 6 +71:5! 4 19! +1:5 Kmin(!) e =9:5! 6 +71! 4 19:5! +1 Kmax(!) o =! 6 +31:5! 4 111! +76:5 K o (!) min =!6 +31! 4 111:5! +76: We can verify the interlacing property of these polynomials (see Figure 57) The illustration in Figure 57shows how all polynomials with even parts bounded by Kmax(!) e and Kmin e (!) and odd parts bounded by Ko max(!) and Kmin o (!) on the imaginary axis satisfy the interlacing property when the Kharitonov polynomials

15 Sec 54 INTERLACING AND IMAGE SET INTERPRETATION Kmax o (!)! e rad/sec K o min (!) B 3 B 4 Kmin (!) B 5 B 1 B K e max (!) Figure 57 Interlacing property of an interval polynomial (Example 54) are stable Figure 57 also shows that the interlacing property for a single stable polynomial corresponding to a point in coecient space generalizes to the box 1 of stable polynomials as the requirement of \interlacing" of the odd and even \tubes" This interpretation is useful; for instance it can be used to show that for polynomials of order less than six, fewer than four Kharitonov polynomials need to be tested for robust stability (see Exercise 54) Image Set Interpretation It is instructive to interpret Kharitonov's Theorem in terms of the evolution of the complex plane image of I(s) evaluated along the imaginary axis Let I(j!) denote the set of complex numbers (j!) obtained by letting the coecients of (s) range over 1: I(j!) :=f(j!) : 1g : Now it follows from the relations in (59) and (51) that I(j!) is a rectangle in the complex plane with the corners K 1 (j!), K (j!), K 3 (j!), K 4 (j!) corresponding to

16 38 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 the Kharitonov polynomials evaluated at s = j! This is shown in Figure 58 As! runs from to 1 the rectangle I(j!) varies in shape size and location but its sides always remain parallel to the real and imaginary axes of the complex plane We illustrate this by using a numerical example Example 55 Consider the interval polynomial of Example 54 The image set of this family is calculated for various frequencies These frequency dependent rectangles are shown in Figure K K 4 Imag 5 K 1 K 3 %! Real Figure 58 Image sets of interval polynomial (Kharitonov boxes) (Example 55) 541 Two Parameter Representation of Interval Polynomials The observation that I(j!) is an axis parallel rectangle with the K i (j!), i =1; ; 3; 4 as corners motivates us to introduce a reduced family I R (s) I(s) which generates the image set I(j!) atevery! Let (s) denote the center polynomial of the family I(s): (s) := x + y + x 1 + y 1 s x n + y n s n

17 Sec 54 INTERLACING AND IMAGE SET INTERPRETATION 39 and introduce the even and odd polynomials: We dene e (s) :=K even max (s) K even min (s) (51) o (s) :=Kmax(s) odd K odd min (s): (5) I R (s) :=f(s) = (s)+ 1 e (s)+ o (s) :j i j1;i=1; g It is easy to see that I R (s) I(s) but I(j!) =I R (j!); for all! : This shows that the n + 1-parameter interval polynomial family I(s) can always be replaced by the two-parameter testing family I R (s) as far as any frequency evaluations are concerned since they both generate the same image at each frequency We emphasize that this kind of parameter reduction based on properties of the image set holds in more general cases, ie even when the family under consideration is not interval and the stability region is not the left half plane Of course in the interval Hurwitz case, Kharitonov's Theorem shows us a further reduction of the testing family to the four vertices K i (s) We show next how this can be deduced from the behaviour of the image set 54 Image Set Based Proof of Kharitonov's Theorem We give an alternative proof of Kharitonov's Theorem based on analysis of the image set Suppose that the family I(s) is of degree n and contains at least one stable polynomialthen stability of the family I(s) can be ascertained by verifying that no polynomial in the family has a root on the imaginary axis This follows immediately from the Boundary Crossing Theorem of Chapter 1 Indeed if some element of I(s) has an unstable root then there must also exist a frequency! 3 and a polynomial with a root at s = j! 3 The case! 3 = is ruled out since this would contradict the requirement that Kmax e () and Ke min () are of the same sign Thus it is only necessary to check that the rectangle I(j! 3 ) excludes the origin of the complex plane for every! 3 > Suppose that the Kharitonov polynomials are stable By the monotonic phase property of Hurwitz polynomials it follows that the corners K 1 (j!), K (j!), K 3 (j!), K 4 (j!) ofi(j!) start on the positive real axis (say), turn strictly counterclockwise around the origin and do not pass through it as! runs from to 1 Now suppose by contradiction that I(j! 3 ) for some! 3 > Since I(j!) moves continuously with respect to! and the origin lies outside of I() it follows that there exists!! 3 for which the origin just begins to enter the set I(j! ) We now consider this limiting situation in which the origin lies on the boundary of I(j! ) and is just about to enter this set as! increases from! This is depicted in Figure 59 The origin can lie on one of the four sides of the image set rectangle, say AB The reader can easily verify that in each of these cases

18 4 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 the entry of the origin implies that the phase angle (argument) of one of the corners, A or B on the side through which the entry takes place, decreases with increasing! at! =! Since the corners correspond to Kharitonov polynomials which are Hurwitz stable, we have a contradiction with the monotonic phase increase property of Hurwitz polynomials Imag 6 Imag 6! s =) B - Real s s # + # A B - Real s! A Imag 6 Imag 6 B s A " " s * B s - Real (= - Real A s Figure 59 Alternative proof of Kharitonov's Theorem

19 Sec 55 EXTREMAL PROPERTIES OF THE KHARITONOV POLYNOMIALS Image Set Edge Generators and Exposed Edges The interval family I(s), or equivalently, the coecient set 1, is a polytope and therefore, as discussed in Chapter 4, its stability is equivalent to that of its exposed edges There are in general (n +1) n+1 such exposed edges However from the image set arguments given above, it is clear that stability of the family I(s) is, in fact, also equivalent to that of the four polynomial segments [K 1 (s);k (s)], [K 1 (s);k 3 (s)], [K (s);k 4 (s)] and [K 3 (s);k 4 (s)] This follows from the previous continuity arguments and the fact that these polynomial segments generate the boundary of the image set I(j!) for each! We now observe that each of the differences K (s) K 1 (s),k 3 (s) K 1 (s),k 4 (s) K (s) and K 4 (s) K 3 (s) is either an even or an odd polynomial It follows then from the Vertex Lemma of Chapter that these segments are Hurwitz stable if and only if the endpoints K 1 (s);k (s);k 3 (s) and K 4 (s) are These arguments serve as yet another proof of Kharitonov's Theorem They serve to highlight 1) the important fact that it is only necessary to check stability of that subset of polynomials which generate the boundary of the image set and ) the role of the Vertex Lemma in reducing the stability tests to that of xed polynomials 55 EXTREMAL PROPERTIES OF THE KHARITONOVPOLY- NOMIALS In this section we derive some useful extremal properties of the Kharitonov polynomials Suppose that we have proved the stability of the family of polynomials (s) = + 1 s + s n s n ; with coecients in the box 1 =[x ;y ] [x 1 ;y 1 ] 111[x n ;y n ]: Each polynomial in the family is stable A natural question that arises now is the following: What point in 1 is closest to instability? The stability margin of this point is in a sense the worst case stability margin of the interval system It turns out that a precise answer to this question can be given in terms of the parametric stability margin as well as in terms of the gain margins of an associated interval system We rst deal with the extremal parametric stability margin problem 551 Extremal Parametric Stability Margin Property We consider a stable interval polynomial family It is therefore possible to associate with each polynomial of the family the largest stability ball centered around it Write =[ ; 1 ; 111; n ]; and regard asapointinir n+1 Let kk p denote the p norm in IR n+1 and let this be associated with (s) The set of polynomials which are unstable of degree n or

20 4 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 of degree less than n is denoted by U Then the radius of the stability ball centered at is () = inf k uk p : uu We thus dene a mapping from 1 to the set of all positive real numbers: 1! R + nfg (s)! (): A natural question to ask is the following: Is there a point in 1 which is the nearest to instability? Or stated in terms of functions: Has the function a minimum and is there a precise point in 1 where it is reached? The answer to that question is given in the following theorem In the discussion to follow we drop the subscript p from the norm since the result holds for any norm chosen Theorem 54 (Extremal property of the Kharitonov polynomials) The function 1! R + nfg (s)! () has a minimum which is reached at one of the four Kharitonov polynomials associated with 1: Proof Let K i (s), i =1; ; 3; 4 denote the four Kharitonov polynomials Consider the four radii associated with these four extreme polynomials, and let us assume for example that (K 1 ) = min (K 1 );(K );(K 3 );(K 4 ) 3 : (5:3) Let us nowsuppose, by way of contradiction, that some polynomial (s) in the box is such that () <(K 1 ): (5:4) For convenience we will denote () by, and (K 1 )by 1 By denition, there is at least one polynomial situated on the hypersphere S((s); ) which is unstable or of degree less than n Let (s) be such a polynomial Since (s) isons((s); ), there exists =[ ; 1 ; 111; n ] with kk =1 such that (s) = + +( )s ( n + n )s n ; (5:5) (, 1, 111, n can be positive or non positive here) But by (53), 1 is the smallest of the four extreme radii and by (54) is less than 1 As a consequence, the four newpolynomials n(s) 1 =(x j j )+(x 1 j 1 j )s +(y + j j )s +(y 3 + j 3 j )s (s) =(x n j j )+(y 1 + j 1 j )s +(y + j j )s +(x 3 j 3 j )s n(s) 3 =(y + j j )+(x 1 j 1 j )s +(x j j )s +(y 3 + j 3 j )s n(s) 4 =(y + j j )+(y 1 + j 1 j )s +(x j j )s +(x 3 j 3 j )s

21 Sec 55 EXTREMAL PROPERTIES OF THE KHARITONOV POLYNOMIALS 43 are all stable because k i n K i k = < i ; i =1; ; 3; 4: By applying Kharitonov's Theorem, we thus conclude that the new box 1 n =[x j j ; y + j j ] 111[x n j n j ; y n + j n j ] (5:6) contains only stable polynomials of degree n The contradiction now clearly follows from the fact that (s) in (55) certainly belongs to 1 n, and yet it is unstable or of degree less than n, and this proves the theorem The above result tells us that, over the entire box, one is closest to instability at one of the Kharitonov corners, say K i (s) It is clear that if we take the box 1 n, constructed in the above proof, (56) and replace by (K i ), the resulting box is larger than the original box 1: This fact can be used to develop an algorithm that enlarges the stability box to its maximum limit We leave the details to the reader but give an illustrative example Example 56 Consider the system given in Example 5: G(s) = s + 1 s + s 3 ( 6 s s + 4 s + 3 ) with the coecients being bounded as follows: [3; 4]; 1 [6;7]; [45; 5]; 3 [4; 3]; 4 [7;8]; 5 [1; 14]; 6 [1; 1]: We wish to verify if the system is robustly stable, and if it is we would like to calculate the smallest value of the stability radius in the space of =[ ; 1 ; ; 3 ; 4 ; 5 ; 6 ] as these coecients range over the uncertainty box The characteristic polynomial of the closed loop system is (s) = 6 s s s s 3 + s + 1 s + : Since all coecients of the polynomial are perturbing independently, we can apply Kharitonov's Theorem This gives us the following four polynomials to check: K 1 (s) = 3 +6s +5s +3s 3 +7s 4 +1s 5 + s 6 K (s) = 3 +7s +5s +4s 3 +7s 4 +14s 5 + s 6 K 3 (s) = 4 +6s +45s +3s 3 +8s 4 +1s 5 + s 6 K 4 (s) = 4 +7s +45s +4s 3 +8s 4 +14s 5 + s 6 : Since all four Kharitonov polynomials are Hurwitz, we proceed to calculate the worst case stability margin

22 44 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 From the result established above,theorem 54 we know that this occurs at one of the Kharitonov vertices and thus it suces to determine the stability radius at these four vertices The stability radius will be determined using a weighted `p norm In other words we want to nd the largest value of p such that the closed loop system remains stable for all satisfying 8 < : [ ;111; 6 ]: " X 6 k k p # 1 p k k= p () where k represent the coecients corresponding to the center (a Kharitonov vertex) It is assumed that [ ; 1 ; ; 3 ; 4 ; 5 ; 6 ]=[43; 33; 5; 15; 5; 1:5; 1] We can compute the stability radius by simply applying the techniques of Chapter 3 (Tsypkin-Polyak Locus, for example) to each of these xed Kharitonov polynomials and taking the minimum value of the stability margin We illustrate this calculation using two dierent norm measures, corresponding to p = and p = 1 9 = ; zk 4(!)! x x z K (!) Imag zk 1(!)! z K 3(!) Real Figure 51 ` stability margin (Example 56)

23 Sec 55 EXTREMAL PROPERTIES OF THE KHARITONOV POLYNOMIALS 45 Figure 51 shows the four Tsypkin-Polyak loci corresponding to the four Kharitonov polynomials The radius of the inscribed circle indicates the minimum weighted ` stability margin in coecient space Figure 511 shows the extremal weighted `1 stability margin From these gures, we have () =1 and 1 () =:4953: zk (!)! x x 4 Imag zk 4(!) - zk 3(!) -4 zk 1(!)! Real Figure 511 `1 stability margin (Example 56) 55 Extremal Gain Margin for Interval Systems Let us now consider the standard unity feedback control system shown below in Figure 51 We assume that the system represented by G(s) contains parameter uncertainty In particular let us assume that G(s) is a proper transfer function which is a ratio of polynomials n(s) and d(s) the coecients of which vary in independent intervals Thus the polynomials n(s) and d(s) vary in respective independent interval polynomial families N (s) and D(s) respectively We refer to such a family of

24 46 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 p - j x - G(s) s Figure 51 A feedback system systems as an interval system Let G(s) := G(s) = n(s) : n(s) N(s); d(s) d(s) D(s) : represent the interval family of systems in which the open loop transfer function lies We assume that the closed loop system containing the interval family G(s) is robustly stable In other words we assume that the characteristic polynomial of the closed loop system given by 5(s) =d(s)+n(s) is of invariant degree n and is Hurwitz for all (n(s),d(s))(n (s) D(s)) Let 5(s) =f5(s) =n(s) +d(s) : n(s) N(s); d(s) D(s)g : Then robust stability means that every polynomial in 5(s) is Hurwitz and of degree n This can in fact be veried constructively Let KN i (s);i =1; ; 3; 4 and K j D (s);j =1; ; 3; 4 denote the Kharitonov polynomials associated with N (s) and D(s) respectively Now introduce the positive set of Kharitonov systems G + K (s) associated with the interval family G(s) as follows: G + K (s) := K i N (s) K i D (s) : i =1; ; 3;4 : Theorem 55 The closed loop system of Figure 51 containing the interval plant G(s) is robustly stable if and only if each of the positive Kharitonov systems in G + K (s) is stable Proof We need to verify that 5(s) remains of degree n and Hurwitz for all (n(s);d(s)) N(s) D(s) It follows from the assumption of independence of the families N (s) and D(s) that 5(s) is itself an interval polynomial family It is easy to check that the Kharitonov polynomials of 5(s) are the four KN i (s) + KD i (s), i = 1; ; 3;4 Thus the family 5(s) is stable if and only if the subset KN i (s) +Ki D (s), i =1; ; 3;4 are all stable and of degree n The latter in turn is equivalent to the stability of the feedback systems obtained by replacing G(s) by each element of G + K (s)

25 Sec 56 ROBUST STATE-FEEDBACK STABILIZATION 47 In classical control gain margin is commonly regarded as a useful measure of robust stability Suppose the closed system is stable The gain margin at the loop breaking point p is dened to be the largest value k 3 of k 1 for which closed loop stability is preserved with G(s) replaced by kg(s) for all k [1;k 3 ) Suppose now that we have veried the robust stability of the interval family of systems G(s) The next question that is of interest is: What is the gain margin of the system at the loop breaking point p? To be more precise we need to ask: What is the worst case gain margin of the system at the point p as G(s) ranges over G(s)? An exact answer to this question can be given as follows Theorem 56 The worst case gain margin of the system at the point p over the family G(s) is the minimum gain margin corresponding to the positive Kharitonov systems G + K (s) Proof Consider the characteristic polynomial 5(s) = d(s)+ kn(s) corresponding to the open loop system kg(s) For each xed value of k this is an interval family For positive k the Kharitonov polynomials of this family are KD i (s)+kki N (s);i = 1; ; 3; 4: Therefore the minimum value of the gain margin over the set G(s) isin fact attained over the subset G + (s) K Remark 56 A similar result can be stated for the case of a positive feedback system by introducing the set of negative Kharitonov systems (see Exercise 59) 56 ROBUST STATE-FEEDBACK STABILIZATION In this section we give an application of Kharitonov's Theorem to robust stabilization We consider the following problem: Suppose that you are given a set of n nominal parameters 8 a ;a 1 ; n19 111;a ; together with a set of prescribed uncertainty ranges: 1a,1a 1, 111,1a n1, and that you consider the family I (s) of monic polynomials, where (s) = + 1 s + s n1s n1 + s n ; a 1a ;a + 1a ; 111; n1 a n1 1a n1 ;a n1 + 1a n1 To avoid trivial cases assume that the family I (s) contains at least one unstable polynomial Suppose now that you can use a vector of n free parameters k =(k ;k 1 ; 111;k n1); :

26 48 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 to transform the family I (s) into the family I k (s) described by: (s) =( + k )+( 1 + k 1 )s +( + k )s ( n1 + k n1)s n1 + s n : The problem of interest, then, is the following: Given 1a,1a 1, 111,1a n1 the perturbation ranges xed a priori, nd, if possible, a vector k so that the new family of polynomials I k (s) is entirely Hurwitz stable This problem arises, for example, when one applies a state-feedback control to a single input system where the matrices A, b are in controllable companion form and the coecients of the characteristic polynomial of A are subject to bounded perturbations The answer to this problem is always armative and is precisely given in Theorem 57 Before stating it, however, we need to prove the following lemma Lemma 53 Let n be a positive integer and let P (s) be a stable polynomial of degree n 1: P (s) =p + p 1 s p n1s n1 ; with all p i > : Then there exists > such that: Q(s) =P (s)+p n s n = p + p 1 s p n1s n1 + p n s n ; is stable if and only if: p n [;): Proof To be absolutely rigorous there should be four dierent proofs depending on whether n is of the form 4r or 4r + 1 or 4r + or 4r +3: We will give the proof of this lemma when n is of the form 4r and one can check that only slight changes are needed if n is of the form 4r + j; j =1; ; 3 If n =4r, r>; we can write P (s) =p + p 1 s p 4r1s 4r1 ; and the even and odd parts of P (s) are given by: Let us also dene P even (s) =p + p s p 4rs 4r ; P odd (s) =p 1 s + p 3 s p 4r1s 4r1 : P e (!) :=P even (j!) =p p! + p 4! 4 111p 4r! 4r ; P o (!) := P odd(j!) = p 1 p 3! + p 5! 4 111p 4r1! 4r : j! P (s) being stable, we know by the Hermite-Biehler Theorem that P e (!) has precisely r 1 positive roots! e;1,! e;, 111,! e;r1, that P o (!) has also r 1 positive roots! o;1,! o;, 111,! o;r1, and that these roots interlace in the following manner: <! e;1 <! o;1 <! e; <! o; < 111<! e;r1 <! o;r1:

27 Sec 56 ROBUST STATE-FEEDBACK STABILIZATION 49 It can be also checked that, P e (! o;j ) < if and only if j is odd, and P e (! o;j ) > if and only if j is even, that is, P e (! o;1) < ; P e (! o;) > ; 111; P e (! o;r) > ; P e (! o;r1) < : (5:7) Let us denote = min j odd P e (! o;j) (! o;j ) 4r : (5:8) By (57) we know that is positive We can now prove the following: Q(s) =P (s)+p 4r s 4r is stable if and only if p 4r [;): Q(s) is certainly stable when p 4r = Let us now suppose that Q o (!) and Q e (!) are given by <p 4r <: (5:9) Q o (!) =P o (!) =p 1 p 3! + p 5! 4 111p 4r1! 4r1 ; Q e (!) =P e (!)+p 4r! 4r = p p! + p 4! 4 111p 4r! 4r + p 4r! 4r : We are going to show that Q e (!) and Q o (!) satisfy the Hermite-Biehler Theorem provided that p 4r remains within the bounds dened by (59) First we know the roots of Q o (!) =P o (!) Then we have that Q e () = p >, and also Q e (! o;1) =P e (! o;1)+p 4r (! o;1) 4r : But, by (58) and (59) we have Thus Q e (! o;1) < Then we have Q e (! o;1) <P e (! o;1) P e (! o;1) (! o;1) (! o;1) 4r : 4r {z } = Q e (! o;) =P e (! o;)+p 4r (! o;) 4r : But by (57), we know that P e (! o;) >, and therefore we also have Q e (! o;) > : Pursuing the same reasoning we could prove in exactly the same way that the following inequalities hold Q e () > ; Q e (! o;1) < ; Q e (! o;) > ; 111;Q e (! o;r) > ; Q e (! o;r1) < : (5:3)

28 5 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 From this we conclude that Q e (!) has precisely r 1 roots in the open interval (,! o;r1), namely! e;1 ;! e; ; 111;! e;r1 ; and that these roots interlace with the roots of Q o (!), <! e;1 <! o;1 <! e; <! o; < 111<! e;r1 <! o;r1: (5:31) Moreover, we see in (53) that Q e (! o;r1) < ; and since p 4r >, we also obviously have Q e (+1) > : Therefore Q e (!) has a nal positive root! e;r which satises! o;r1 <! e;r : (5:3) From (531) and (53) we conclude that Q o (!) and Q e (!) satisfy the Hermite- Biehler Theorem and therefore Q(s) is stable To complete the proof of this lemma, notice that Q(s) is obviously unstable if p 4r < since we have assumed that all the p i are positive Moreover it can be shown that for p 4r =, the polynomial P (s)+s 4r has a pure imaginary root and therefore is unstable Now, it is impossible that P (s) +p 4r s 4r be stable for some p 4r >, because otherwise we could use Kharitonov's Theorem and say, P (s)+ s4r and P (s)+p 4r s 4r both stable =) P (s)+s 4r stable ; which would be a contradiction This completes the proof of the theorem when n =4r For the sake of completeness, let us just make precise that in general we have, P e if n =4r; = min (! o;j) ; j odd (! o;j ) 4r P o if n =4r +1; = min (! e;j) ; j even (! o;j ) 4r+1 P e if n =4r +; = min (! o;j) ; j even (! o;j ) 4r+ if n =4r +3; = min j odd P o (!e;j) (! o;j ) 4r+3 : The details of the proof for the other cases are omitted We can now enunciate the following theorem to answer the question raised at the beginning of this section

29 Sec 56 ROBUST STATE-FEEDBACK STABILIZATION 51 Theorem 57 For any set of nominal parameters fa ;a 1 ; 111;a n1g, and for any set of positive numbers 1a, 1a 1, 111, 1a n1, it is possible to nd a vector k such that the entire family I k (s) is stable Proof The proof is constructive Step 1: Take any stable polynomial R(s) of degree n 1 Let (R(1)) be the radius of the largest stability hypersphere around R(s) It can be checked from the formulas of Chapter 3,that for any positive real number,we have (R(1)) = (R(1)): Thus it is possible to nd a positive real such that if P (s) =R(s), s 1a (P (1)) > + 1a a n1 : (5:33) Denote P (s) =p + p 1 s + p s p n1s n1 ; and consider the four following Kharitonov polynomials of degree n 1: P 1 (s) = p 1a + p 1 1a 1 s + p + 1a s + 111; P (s) = p 1a + p 1 + 1a 1 s + p + 1a s + 111; P 3 (s) = p + 1a + p 1 1a 1 s + p 1a s + 111; (534) P 4 (s) = p + 1a + p 1 + 1a 1 s + p 1a s + 111: We conclude that these four polynomials are stable since kp i (s) P (s)k <(P (s)): Step : Now,applying Lemma 53,we know that we can nd four positive numbers 1,, 3, 4,such that P j (s)+p n s n is stable for p n < j ; j =1; ; 3;4: Let us select a single positive number such that the polynomials, P j (s)+s n (5:35) are all stable If can be chosen to be equal to 1 (that is if the four j are greater than 1) then we do choose = 1; otherwise we multiply everything by 1 which is greater than 1 and we know from (535) that the four polynomials K j (s) = 1 P j (s)+s n ;

30 5 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 are stable But the four polynomials K j (s) are nothing but the four Kharitonov polynomials associated with the family of polynomials where (s) = + 1 s n1s n1 + s n ; 1 p 1 1a ; 1 p + 1 1a ; ; n1 p n1 1 1a n1 1 ; p n a n1 and therefore this family is entirely stable Step 3: It suces now to chose the vector k such that ; k i + a i = 1 p i; for i =1; 111;n 1: Remark 57 It is clear that in step 1 one can determine the largest box around R(1) with sides proportional to 1a i The dimensions of such a box are also enlarged by the factor when R(1) is replaced by R(1) This change does not aect the remaining steps of the proof Example 57 Suppose that our nominal polynomial is s 6 s 5 +s 4 3s 3 +s + s +1; that is (a ;a 1 ;a ;a 3 ;a 4 ;a 5 )=(1; 1; ; 3;; 1): And suppose that we want to handle the following set of uncertainty ranges: 1a =3; 1a 1 =5; 1a =; 1a 3 =1; 1a 4 =7; 1a 5 =5: Step 1: Consider the following stable polynomial of degree 5 R(s) =(s +1) 5 =1+5s +1s +1s 3 +5s 4 + s 5 : The calculation of (R(1)) gives: (R(1)) = 1 On the other hand we have s 1a + 1a a n1 =5:31: Taking therefore = 6, we have that P (s) =6+3s +6s +6s 3 +3s 4 +6s 5 ;

31 Sec 57 POLYNOMIAL FUNCTIONS OF INTERVAL POLYNOMIALS 53 has a radius (P (1)) = 6 that is greater than 5:31 The four polynomials P j (s) are given by P 1 (s) =4:5+7:5s +61s +6:5s 3 +6:5s 4 +3:5s 5 ; P (s) =4:5+3:5s +61s +59:5s 3 +6:5s 4 +8:5s 5 ; P 3 (s) =7:5+7:5s +59s +6:5s 3 +33:5s 4 +3:5s 5 ; P 4 (s) =7:5+3:5s +59s +59:5s 3 +33:5s 4 +8:5s 5 : Step : The application of Lemma 53 gives the following values 1 ' 1:36; ' :667; 3 ' 1:784; 4 ' 3:81; and therefore we can chose = 1, so that the four polynomials are stable K 1 (s) =4:5+7:5s +61s +6:5s 3 +6:5s 4 +3:5s 5 + s 6 ; K (s) =4:5+3:5s +61s +59:5s 3 +6:5s 4 +8:5s 5 + s 6 ; K 3 (s) =7:5+7:5s +59s +6:5s 3 +33:5s 4 +3:5s 5 + s 6 ; K 4 (s) =7:5+3:5s +59s +59:5s 3 +33:5s 4 +8:5s 5 + s 6 ; Step 3: We just have to take k = p a =5; k1 = p1 a1 =9; k = p a =58; k3 = p3 a3 =63; k4 = p4 a4 =8; k5 = p5 a5 =7: 57 POLYNOMIAL FUNCTIONS OF INTERVAL POLYNOMI- ALS In this section we extend the family of polynomials to which Kharitonov's Theorem applies Consider the real interval polynomial and a given polynomial I(s) = 8 a(s) :aj [a j ;a+ j ]; j =; 1; ; 111;n9 ; (5:36) We generate the polynomial family '(z) = + 1z mz m : (5:37) '(I(s)) = f'(a(s)) : a(s) I(s)g (5:38)

32 54 INTERVAL POLYNOMIALS: KHARITONOV'S THEOREM Ch 5 which consists of polynomials of the form (s)+1a(s)+a (s)+111+ m a m (s) (5:39) where a(s) I(s) In the following we answer the question: Under what conditions is the family '(I(s)) Hurwitz stable? We note that the family '(I(s)) is in general neither interval nor polytopic Moreover, even though the image set I(j!) is a rectangle the image set of '(I(j!)) is, in general a very complicated set Therefore it is not expected that a vertex testing set or an edge testing will be available However the result given below shows that once again it suces to test the stability of four polynomials Let K 1 (s);k (s);k 3 (s);k 4 (s) denote the Kharitonov polynomials associated with the family I(s) We begin with a preliminary observation Lemma 54 Given the real interval polynomial I(s) and a complex number z the Hurwitz stability of the family I(s) z = fa(s) z : a(s) I(s)g is equivalent to the Hurwitz stability of the polynomials K j (s) z, j =1; ; 3;4 The proof of this lemma follows easily from analysis of the image set at s = j! of I(s) z and is omitted Stability Domains For a given domain of the complex plane let us say that a polynomial is -stable if all its roots lie in the domain Consider a polynomial a k (s) of degree n The plot z = a k (j!),! (1; +1), partitions the complex plane into a nite number of open disjoint domains With each such domain 3 we associate the integer n3, the number of the roots of the polynomial a k (s) z in the left half plane, where z is taken from 3 This number is independent of the particular choice of z in the domain and there is at most one domain 3 k for which n3 = n Let us associate this domain 3 k with a k (s) and call it the stability domain If it so happens that there is no domain for which n3 = n we set 3 k = ; Dene for each polynomial K j (s), the associated stability domain 3 j, j = 1; ; 3; 4 and let 3 := \ 4 k=13 k : Theorem 58 Suppose that 3 6= ;: The polynomial family '(I(s)) is Hurwitz stable if and only if '(z) is 3-stable, that is, all the roots of '(z) lie in the domain 3